Shock formation in nonlinear transport equation

Question

The solution implies that $u_0(0-)$ is equal to $\alpha$ . However the question states that $u_0(x)=\alpha$ for $x>0$. Wouldn't $u_0(0-)$ be equal to the "smooth function for $x<0$"? I have underlined the relevant parts in orange.

Answer 1

Think of the purpose of the example: to show shock formation. Shock is a discontinuity of velocity $u$. So to demonstrate how it forms, we should begin with continuous velocity field; otherwise the shock is already there at time $0$, and we just study its propagation.

Yes, the authors consider that "smooth function for $x<0$" is has limit $\alpha$ as $x\to 0-$, which makes the function $u_0$ continuous. They neglected to state this explicitly, but I think the intent is clear. In the first sentence, I would replace "piecewise continuous" by "piecewise smooth".